Ordinal numbers, transfinite: numbers of a type defined by Georg Cantor that are larger than any finite number. Transfinite ordinal numbers are represented by the letter Omega, which is the last letter of the Greek alphabet.

How to conceive of such a number? In his novel White Light, the author Rudy Rucker* describes the mountain On, whose height is more than infinite. The mountain consists of an infinite series of cliffs one on top of another. Even the most infinite cliff is followed by yet another infinite series of cliffs, and this is repeated infinitely many times.

Climbing Infinite Cliffs

From time immemorial, rock climbers have tried to climb the mountain On. On each cliff that they climbed, they painted the corresponding cliff number with weatherproof paint for their orientation. You would think that hardly any of them has ever reached the top of this more-than-infinite mountain. However, in doing so you would greatly underestimate the climbing skills of our On mountaineers, for as every climber knows, the more you climb the more expert you become at doing it. The learning effect resulting from the overcoming of each cliff is so great that the climber can double his or her speed afterwards. Thus, if you needed an hour to master the first cliff, you will make the second one in only half an hour, and the third one in only quarter of an hour (assuming that the cliffs are equally high and equally challenging). As a result, the total climbing time in hours corresponds to the following sum:

As you can read in the article on sequences, the sum of this infinite sequence amounts to the finite number 2. Hence, it takes our climber only two hours to reach the cliff with the number ω — only to discover that she has still not reached the top of the mountain. More cliffs await her — many, many more. And these, too, have already been mastered by other climbers before her. She can see this from the numbers ω+1, ω+2, and so forth, that have been painted on the cliffs.**

The Frustrated Climber

Since Omega is larger than any finite number, after a while the numbers on the cliffs become infinitely large; these are Cantor's transfinite ordinal numbers. An ordinal numberis a member of an ordered set of numbers such that for any two members of that set, it can be determined which one is larger. The set of natural numbers is one such ordered set. The set of cliffs on the mountain On is also ordered in this way, since for any two of the cliffs it can be easily decided which is higher. The set of all points in a plane, by contrast, is not ordered, since there are no differences in size or quantity among these points.***

A set of numbers that can be represented on a number ray — that is, along a line — is obviously ordered, since here we can decide for each pair of numbers that the number located to the right on the line is larger. But what happens if both numbers are infinitely large?

Cantor approached this probem by simply continuing to count beyond the finite numbers. He marked the smallest number on the number ray that is larger than any finite number with the symbol ω. Of course, from here it is still a long way to the end of the number ray. Therefore, we can continue counting from ω with ω+1, ω+2, and so on, until we get to ω+ω = ω∙2:

1, 2,... ω, ω+1, ω+2,... ω∙2, ω∙2+1, ω∙2+2,...

In the same manner, we can define infinitely many lines on the number ray, each of which has a length of ω. If we place them side by side, these lines constitute an infinitely large plane — a plane composed of ω lines with ω numbers each. Thus, the plane comprises ω∙ω = ω2 ordered numbers.

Strange Calculation Rules

Like the infinite cardinals, the ω numbers are subject to special calculation rules. ω+1 is larger than ω, yet 1+ω = ω! The reason is that since ω is larger than any finite number, ω is also larger than 1 plus a finite number (which itself is, obviously, another finite number). Likewise, we can easily figure out that 2∙ω = ω but at the same time ω∙2 > ω. Commutativity (permutability of the order of factors) of addition and multiplication is valid only for finite numbers.

However, we are far from finished with infinities. Above this plane of infinite lines we can define another plane made up of just as many lines. The first line of the second plane begins with the transfinite ordinal number ω2.

 ω2, ω2+1, ω2+2,. ω2+ω, ω2+ω+1, ω2+ω+2,.

Again, a third plane can be defined above the second one whose first line begins with the numerical sequence ω2∙2, ω2∙2+1, ω2∙2+2,.. Above that we can define another plane, and so forth until we obtain an infinitely large cube:

1, 2,... ω, ω+1, ω+2,... ω∙2,... ω2, ω2+1, ω2+2,...

This hypothetical cube contains ω3 numbers. But we are still not finished, for we can add to this cube a second cube, a third one, a fourth one, and so on. At that point we are within the fourth dimension and can start all over by first constructing a line, then a plane, and finally a cube of infinitely many cubes. A hyper-line of this kind contains ω4, the corresponding hyper-plane ω5, and the corresponding hyper-cube ω6 numbers.

Infinite-Dimensional Planes

Two thousand years before Cantor, another mathematician had already used the same method to construct a system of extremely large numbers; this was Archimedes with his calculation of the number of grains of sand. To be sure, Cantor went further. He repeated the above operation with infinitely large cubes serving as the basic units of ever new lines, planes, and cubes an infinite number of times. The number at which he eventually arrived was ωω, which is the precise quantity of numbers in an infinite-dimensional space of infinitely long segments of numbers. And this still does not complete the set of transfinite ordinal numbers:

1, 2,... ω,. ω∙2,... ω2, ω2+1, ω2+2,... ωω, ωω+1, ωω+2,...

Is there any end in sight? If you look at the sequence of omegas, you may notice that ω is subjected at regular intervals to a higher mathematical operation involving ω itself:

ω,... ω+ω,... ω∙ω,... ωω,...

Omega is first added to itself, then multiplied by itself, and finally raised to the power of itself. Each of these operations corresponds to ω times the next lower operation. Do you now get a sense of what comes after exponentiation? Let's call it "hyperization". Thus, omega hyper 4 would mean that omega is raised three times to the power of itself:

Our infinite-dimensional space of number segments thus ends with the number omega hyper omega or ω. But does it really end? No, in fact it does not. Though ω is an incredibly large number, there is still a larger one, namely ω+1. It's time we tried to bring the omegas under control by means of the next higher mathematical operation, namely, hyper hyperization. Omega hyper hyper 4 hyperizes omega three times with itself:

However, as you will have guessed, even after mastering cliff no. ωHHHHHHHHHHHHHHω our climber has still not reached the top of mountain On. Indeed, no hyper-wise calculating operation will enable us to identify the number that would be painted above the highest cliff on the summit of that mountain. We therefore have to content ourselves with one symbol, the Great Omega, as the cross on the summit of our mountain On:

And here we have reached the end. There is no Ω+1, because Ω by itself is not a number. At best, we can say that Ω stands for the concept of the never-ending continuance of counting ordinal numbers beyond all limits, hence for the absolutely infinite.****


* Our recommended reading list includes a book by the same author about infinity that is worth reading.

** To be sure, our climber has become so fit after doing ω cliffs that she should be able to climb the succeeding cliffs in an infinitely short period of time. Nonetheless, another disappointment will await her upon reaching the summit and enjoying the views. For she will inevitably make out a mountain at the far distant horizon that is much larger than the mountain On. You can read up on the story of this second mountain in the last footnote to this article.

*** Of course, we could in principle define such a difference in size for points (for example, by means of a function), and would thereby obtain an ordered set of points.

**** If you carefully compare the articles on cardinals and ordinal numbers, you should notice something paradoxical. The cardinality of the set of natural numbers is א0. But since the ordinal numbers are countable and all countable sets are of the same cardinality, א0 is the cardinality of all numbers up to ω and therefore of all numbers up to ωω. In the article on cardinals, however, it had been stated that 2א0 > א0. How is this possible, given that the latter set contains as many as ωω elements, which is obviously more than 2ω?

Here again, we encounter our old erroneous intuitions about infinity. The truth is that the set of numbers up to ωω does not contain any more elements than the set of numbers up to ω. Both sets contain the same number of elements. The cardinal 2א0, by contrast, pertains to a far more powerful set. A mountain visible at the distant horizon whose number of cliffs corresponds to the number of elements of a set of cardinality 2א0 would make our mountain On look like a pathetic little mound. To be sure, the cliffs of such a mountain would no longer be countable, and there wouldn't be numbers painted on most of them.

Links Related to the Topic

■ Rudy Rucker
■ Wikipedia: Ordinal Number
■ White Light
■ Infinity: Mathematicians grappling about a concept

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